Group #15674538 — subgroups

Subgroups — #15674538

Order 2944.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625040	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625017	C₂	1	2	yes	maximal
3	#13092179	C₂	1	2	yes	maximal
4	#13092172	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#391	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×Dic₄₆	—	2	1	no	—
8	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₂₃	—	2	1	no	—
11	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
12	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
13	#10625040	—	2	4	no	—
14	C₂₃×#95	—	2	1	no	—
15	C₂₃×#95	—	2	1	no	—
16	#10625017	—	2	4	no	—
17	C₂×Q₈×C₂₃	—	2	1	no	—
18	C₂×Q₈×C₂₃	—	2	2	no	—
19	C₂×Q₈×C₂₃	—	2	2	no	—
20	C₂×Q₈×C₂₃	Q₈	1	2	yes	complemented
21	C₂²×C₄×C₂₃	—	2	1	no	—
22	C₂²×C₄×C₂₃	—	2	1	no	—
23	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₂₃	—	2	2	no	—
27	C₂×Dic₄₆	—	2	4	no	—
28	C₂×Dic₄₆	—	4	2	no	—
29	C₂×Dic₄₆	—	4	2	no	—
30	C₄⋊C₄×C₂₃	—	2	2	no	—
31	C₄²×C₂₃	—	2	2	no	—
32	C₄⋊Dic₂₃	—	2	4	no	—
33	C₄⋊Dic₂₃	—	2	2	no	—
34	C₄⋊Dic₂₃	D₄	1	4	yes	—
35	C₂×C₂₃⋊C₈	—	2	2	no	—
36	C₂×C₂₃⋊C₈	—	2	2	no	—
37	C₂×C₂₃⋊C₈	—	2	2	no	—
38	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₂₃	—	2	4	no	—
40	Q₈×C₂₃	—	4	2	no	—
41	C₂×C₄×C₂₃	—	2	1	no	—
42	C₂×C₄×C₂₃	—	2	1	no	—
43	C₂×C₄×C₂₃	—	2	1	no	—
44	C₂×C₄×C₂₃	—	2	1	no	—
45	C₂×C₄×C₂₃	—	2	2	no	—
46	C₂×C₄×C₂₃	—	4	1	no	—
47	C₂×C₄×C₂₃	—	4	1	no	—
48	C₂×C₄×C₂₃	—	4	1	no	—
49	C₂×C₄×C₂₃	—	4	1	no	—
50	C₂×C₄×C₂₃	—	4	1	no	—
51	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
52	C₂×C₄×C₂₃	Q₁₆	1	2	yes	—
53	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
56	C₂×C₄×C₂₃	#41	1	2	yes	—
57	C₂×Dic₂₃	—	2	4	no	—
58	C₂×Dic₂₃	—	4	2	no	—
59	C₂×Dic₂₃	—	4	2	no	—
60	Dic₄₆	—	4	8	no	—
61	C₂₃⋊C₈	—	4	2	no	—
62	#2024	—	23	1	no	maximal, Sylow
63	C₂²×C₂₃	#120	1	1	yes	characteristic
64	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂²×C₂₃	#124	1	1	yes	characteristic
66	C₂²×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₂²×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₂²×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₂²×C₂₃	#137	1	1	yes	characteristic
70	C₄×C₂₃	—	2	2	no	—
71	C₄×C₂₃	—	2	2	no	—
72	C₄×C₂₃	—	4	2	no	—
73	C₄×C₂₃	—	8	1	no	—
74	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
75	C₄×C₂₃	#124	1	2	yes	—
76	Dic₂₃	—	4	4	no	—
77	C₂×C₄⋊Q₈	—	23	1	no	—
78	C₂×#103	—	23	2	no	—
79	#391	D₂₃	1	1	yes	characteristic
80	#340	—	23	2	no	—
81	#336	—	23	1	no	—
82	C₂×C₂₃	#393	1	1	yes	characteristic
83	C₂×C₂₃	#450	1	1	yes	characteristic
84	C₂×C₂₃	#451	1	1	yes	characteristic
85	C₂×C₂₃	#475	1	1	yes	characteristic
86	C₂×C₂₃	#477	1	1	yes	characteristic
87	C₂×C₂₃	#487	1	1	yes	characteristic
88	C₂×C₂₃	#489	1	1	yes	characteristic
89	C₂²×Q₈	—	46	1	no	—
90	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
91	C₂²×C₈	—	23	2	no	—
92	C₄⋊Q₈	—	46	4	no	—
93	C₂×C₄⋊C₄	—	23	2	no	—
94	C₂×C₄⋊C₄	—	46	1	no	—
95	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
96	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
97	#103	—	46	4	no	—
98	#95	—	2	1	no	—
99	#95	—	2	1	no	—
100	C₂₃	#2024	1	1	yes	characteristic, complemented, Sylow
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	46	4	no	—
105	C₂×Q₈	—	92	2	no	—
106	C₂×Q₈	—	92	2	no	—
107	C₂×Q₈	Dic₄₆	1	2	yes	complemented
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	46	2	no	—
111	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
114	C₂×C₈	—	46	2	no	—
115	C₂×C₈	—	46	2	no	—
116	C₂×C₈	—	46	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	23	4	no	—
119	C₄⋊C₄	—	46	4	no	—
120	C₄⋊C₄	—	46	2	no	—
121	C₄²	—	2	2	no	—
122	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
123	Q₈	—	2	4	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	92	8	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	46	4	no	—
137	C₂×C₄	—	92	2	no	—
138	C₂×C₄	—	92	2	no	—
139	C₂×C₄	#67274	1	2	yes	—
140	C₂×C₄	#67275	1	2	yes	—
141	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
142	C₂×C₄	#67288	1	1	yes	characteristic
143	C₂×C₄	D₄×D₂₃	1	2	yes	—
144	C₂×C₄	#67290	1	2	yes	—
145	C₈	—	92	2	no	—
146	C₂²	#10625095	1	1	yes	characteristic
147	C₂²	#10625107	1	1	yes	characteristic
148	C₂²	#10625108	1	1	yes	characteristic
149	C₂²	C₂×#67274	1	1	yes	characteristic
150	C₂²	C₂×#67275	1	1	yes	characteristic
151	C₂²	#10625120	1	1	yes	characteristic
152	C₂²	#10625122	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	—	8	1	no	—
157	C₄	—	92	4	no	—
158	C₄	#10625049	1	2	yes	—
159	C₄	#10625056	1	2	yes	—
160	C₂	#13092329	1	1	yes	characteristic
161	C₂	#13092567	1	1	yes	characteristic
162	C₂	#13092572	1	1	yes	characteristic
163	C₂	#13092702	1	1	yes	characteristic
164	C₂	#13092712	1	1	yes	characteristic
165	C₂	#13092767	1	1	yes	characteristic
166	C₂	#13092776	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625040

C₂

yes

maximal, characteristic

C₂×#10625017

C₂

yes

maximal

#13092179

C₂

yes

maximal

#13092172

C₂

yes

maximal, characteristic

C₂₃×#391

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₂²

yes

characteristic

C₂²×Dic₄₆

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10625040

—

C₂₃×#95

—

C₂₃×#95

—

#10625017

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

Q₈

yes

complemented

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

Q₁₆

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₂²×C₂₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₃

#124

yes

characteristic

C₂²×C₂₃

C₂×SD₁₆

yes

characteristic

C₂²×C₂₃

C₂×Q₁₆

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

#124

yes

—

Dic₂₃

—

C₂×C₄⋊Q₈

—

C₂×#103

—

#391

D₂₃

yes

characteristic

#340

—

#336

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×Q₈

—

C₂²×Q₈

C₂×D₂₃

yes

characteristic

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₂×C₄²

C₂×D₂₃

yes

—

—

—

100

C₂₃

#2024

yes

characteristic, complemented, Sylow

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

Dic₄₆

yes

complemented

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

112

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

113

C₂²×C₄

C₂²×D₂₃

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄²

—

122

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#67274

yes

—

140

C₂×C₄

#67275

yes

—

141

C₂×C₄

C₂×Dic₄₆

yes

characteristic

142

C₂×C₄

#67288

yes

characteristic

143

C₂×C₄

D₄×D₂₃

yes

—

144

C₂×C₄

#67290

yes

—

145

C₈

—

146

C₂²

#10625095

yes

characteristic

147

C₂²

#10625107

yes

characteristic

148

C₂²

#10625108

yes

characteristic

149

C₂²

C₂×#67274

yes

characteristic

150

C₂²

C₂×#67275

yes

characteristic

151

C₂²

#10625120

yes

characteristic

152

C₂²

#10625122

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#10625049

yes

—

159

C₄

#10625056

yes

—

160

C₂

#13092329

yes

characteristic

161

C₂

#13092567

yes

characteristic

162

C₂

#13092572

yes

characteristic

163

C₂

#13092702

yes

characteristic

164

C₂

#13092712

yes

characteristic

165

C₂

#13092767

yes

characteristic

166

C₂

#13092776

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂²×C₄×C₂₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#391	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
hypercenter	#391	D₂₃
hypocenter	C₂₃	#2024
LCS	G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator, cosocle

C₂²×C₄×C₂₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#391

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

D₂₃

C₂₃

G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
20. C₂×Q₈×C₂₃	Q₈	semidirect
100. C₂₃	#2024	semidirect
107. C₂×Q₈	Dic₄₆	semidirect

H ≅ G/N

type

20. C₂×Q₈×C₂₃

Q₈

semidirect

100. C₂₃

#2024

semidirect

107. C₂×Q₈

Dic₄₆

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#10625040	2
C₂×#10625017	2
#13092179	2
#13092172	2
C₂₃×#391	2
#2024	23

subgroup

index

C₂₃×#391

Subgroups — #15674538

Conjugacy classes of subgroups

Complements

Maximal subgroups